Don’t buy tickets for each ride! Unless you’re in London for just a day (and maybe not even then), paying per ride is exorbitantly expensive. An Oyster card requires a £5 deposit (which you can get back at the end of your trip), and then you can load it with money or, even better, 1, 7, or 30 day TravelCards. These allow you unlimited travel within the zones specified for the purchased amount of time. Super easy. To give you a rough idea how much cheaper this is, a single ride during the day can cost upwards of £6.50, whereas a 1-day TravelCard costs £12.10. Here's a rate table (pdf).

You can get an Oyster card at nearly every Underground station. Once you’ve got your Oyster loaded up, all you need to do is tap it when you enter and leave the system (you’ll see everyone else do it too). Easy.

There are also Visitor Oyster Cards. As TfL explains "It is the cheapest way to pay for single journeys on bus, Tube, tram, DLR, London Overground, Tfl Rail and most National Rail services in London." The "single journey" is the key there. They list £2.90 (instead of £4.90 for a single ticket). They also explain: "If you are in London for a week and plan to use only the central London Travelcard zones on a daily basis then a 7 day Travelcard might be the most cost-effective option. If you plan to stay in London for a shorter time or want to travel to the outskirts of the city, the pay-as-you-go Oyster card cost might be cheaper."

So, less than 7 days, it might be cheaper to get the Visitor Card. Might. Hard to say. Personally, I think just getting a regular Oyster and load a 7-day TravelCard on it (£32.40) is easiest.

It’s worth mentioning that all the big touristy spots are Zone 1, so the basic Zone 1-2 passes will work for most people (though it depends where you’re staying, of course). Heathrow is Zone 6, so that's worth considering, especially since the ultra-fast Heathrow Express costs ~$30 one way (~$50 round trip), though less if you buy in advance.